Calculus of Variations and Geometric Measure Theory

A. Pratelli

On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

created on 21 Oct 2004
modified by pratelli on 25 Sep 2010


Published Paper

Inserted: 21 oct 2004
Last Updated: 25 sep 2010

Journal: Ann. Inst. H. Poincare' Probab. Statist.
Year: 2005


This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the existence of classic transport maps and the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. As a byproduct, we can prove a stronger version of the classic Isomorphism Theorem of measure rings. A preliminary version of the results of this paper is contained in the Ph.D. thesis \cite{Prthesis}.